Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 4

On a fixed horizon $T$, an arbitrage is an admissible trading strategy that is self-financing, has $X_0=0$, and satisfies $X_T\ge 0$ almost surely with $\mathbb{P}(X_T>0)>0$. Here admissibility means that wealth in units of the strictly positive riskless numéraire has a deterministic lower bound throughout the horizon. An equivalent martingale measure $Q$ is equivalent to the physical measure on ${\mathcal{F}}_T$ and makes each discounted risky price ${\tilde{S}}^i=S^i/B$ a martingale.

For a predictable, stochastically integrable risky holding vector $\pi$, the discounted self-financing identity isIndeed, since the riskless account is a finite-variation process, its quadratic covariations vanish, and the Itô formula for $X/B$, combined with $X=\varphi B+\pi \cdot S$ and $dX=\varphi dB+\pi \cdot dS$, gives $d(X/B)=\pi \cdot d(S/B)$.

Use two standard facts from stochastic calculus: a stochastic integral against a continuous local martingale is a local martingale, and a local martingale bounded below by a deterministic constant is a supermartingale. The latter follows by adding the constant to make the process nonnegative, stopping at a localizing sequence, and applying the conditional Fatou lemma. Thus admissible discounted wealth is a $Q$-supermartingale, and an arbitrage would satisfyIt follows that $X_T=0$ $Q$-almost surely, hence also physically almost surely by equivalence, a contradiction. Therefore an equivalent martingale measure rules out arbitrage among admissible self-financing strategies. The lower-bound restriction is essential to this argument; stochastic integrals need not be true martingales for arbitrary unbounded holdings.

The holdings ${\varphi }_t$ and ${\pi }_t^i$ are numbers of shares, so the marked-to-market portfolio value isThe stock prices are ex-dividend prices. Over the trading interval, existing holdings gain ${\varphi }_{t}d{B}_t$ from the riskless account and ${\pi }_t\cdot d{S}_t$ from capital-price changes. In addition, ${\pi }_t^i$ shares of asset $i$ receive ${\pi }_t^i{D}_t^{i}dt$ in cash dividends. If this cash remains in the portfolio and changes of holdings are financed internally, the self-financing portfolio equation isRebalancing does not add a separate source of wealth: the cost of purchasing one holding is paid by selling another or by using the cash account. This is why one must specify the gains equation, rather than infer self-financing solely by differentiating the marked-to-market identity. The dividends are reinvested rather than withdrawn as consumption.

Use the discounted wealth $\tilde{X}=X/B$ and the discounted dividend gainsThe integral uses $ds$: the printed $dt$ inside an integral indexed by $s$ is a differential typo. Since $B$ is a positive finite-variation process, the Itô product rule givesThe bank-account terms cancel using $X_t={\varphi }_t{B}_t+{\pi }_t\cdot S_t$.

If $Y$ is a $Q$-martingale, discounted self-financing wealth is a $Q$-local martingale. The admissible trading strategy lower bound makes it a supermartingale, exactly as in the previous no-arbitrage proof. A zero-cost portfolio with nonnegative terminal wealth must therefore have terminal wealth zero $Q$-almost surely and, by equivalence, physically almost surely. Hence an equivalent martingale measure for discounted dividend gains excludes arbitrage. It is the gains process, not just the ex-dividend price ratio, that must have zero martingale drift.